Theorem ectocl 6173

Description: Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
ectocl.1	|- S = ( B /. R )
ectocl.2	|- ( [ x ] R = A -> ( ph <-> ps ) )
ectocl.3	|- ( x e. B -> ph )

Assertion

Ref	Expression
ectocl	|- ( A e. S -> ps )

Distinct variable groups:   x, A   x, B   x, R   ps, x

Allowed substitution hints:   ph( x)   S( x)

Proof of Theorem ectocl

Step	Hyp	Ref	Expression
1		tru 1247	. 2 |- T.
2		ectocl.1	. . 3 |- S = ( B /. R )
3		ectocl.2	. . 3 |- ( [ x ] R = A -> ( ph <-> ps ) )
4		ectocl.3	. . . 4 |- ( x e. B -> ph )
5	4	adantl 262	. . 3 |- ( ( T. /\ x e. B ) -> ph )
6	2, 3, 5	ectocld 6172	. 2 |- ( ( T. /\ A e. S ) -> ps )
7	1, 6	mpan 400	1 |- ( A e. S -> ps )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   T. wtru 1244   e. wcel 1393   [cec 6104   /.cqs 6105

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-qs 6112

This theorem is referenced by: (None)